# Full text of "Treatise On Analysis Vol-Ii"

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12 UNBOUNDED NORMAL OPERATORS 421 Now let /0 denote the characteristic function of the set {0}, and for n ^ 1 let/n denote the characteristic function of the interval ]!/(« + 1), 1/ri] in R. For each C e R, the series /0(Q +/i(0 + • • • +/„(£) + * * * converges to the value at £ of the characteristic function of the interval [0,1]. For each n 2> 0, let Fn be the image of E under the orthogonal projector/n(£) = PFn (15.10.6). From the preceding discussion and from (15.11.8(iv)) it follows that E is the Hilbert sum of the Fn (n S> 0). Also, in fact, F0 = {0}, because F0 is the kernel Qi the operator B (15.11.6), and by virtue of (15.12.6) the hermitian form (x, y) i-> (B • x \ y) is nondegenerate, that is to say the relation (B*x\x) = Q implies x = 0, and so F0 = {0}. It follows from above that PFn(dom(AO) <= dom(JV) and that PFnN- x = NPFn • x for all x e dom(N). On the other hand, the definition of the functions /„ for n ^ 1 shows that the function gn(£) = C~V»(0 (with &i(0) = 0) is a bounded regulated function on R and therefore universally measurable (7.6.1). Consequently we may write PFn = Bgn(B\ whence JVPFn = NBgn(B). But we have shown (15.12.6) that NB is a continuous operator defined on E, and hence the restriction Nn of N to Fn n dom(N) is a continuous mapping of this subspace into Fn. Since PFrt(dom(7\0) c= Fn n dom(JV)» it follows that Fn n dom(JV) is dense in Fn; also Nn is clearly a closed operator on Fn, hence by (15.12.2) we conclude that Fn c dom(N). Also, since N*N = NN*, we may interchange N and N* in the above discussion, which does not change B; hence the restriction N'n of JV* to Fn is also a continuous operator on this subspace, and is evidently equal to the adjoint N* of Nn. It is now clear that Nn is normal, and the proof of (15.12.8) is complete. (15.12.9) Let T be a closed (not necessarily bounded) operator on E. Generalizing (11.1), we say that £ e C is a regular value of T if the operator T — £7 is a bijective linear mapping of dom(T) onto E, and if the inverse linear mapping JRr(Q : E -> dom( T) is continuous. (15.12.10) For C e C to be a regular value of a closed operator T, it is sufficient that T— U should be an injective linear mapping of dom(T) onto a dense subspace L 0/E, and that the inverse mapping (T — C/)""1 : L-» E should be continuous. For the graph of T— C/ is closed, and so is the graph of (r— C/)"1 (15.12.5). Hence it follows from (15.12.2) that L is closed in E, and there- fore L = E. The complement in C of the set of regular values of T is again called the spectrum of T, and is denoted by Sp(T). If C e Sp(T), it follows from (15.12.10) that there are three possibilities:ciple" is satisfied,